Starting next year the island will cap daily cruise arrivals at 8,000, hoping to curb a flow of day travelers the mayor said is putting too much strain on infrastructure and supplies. The move has rankled cruise operators and business owners, who say such a move would choke off their economic lifeblood and that the onus is on the island to do more to accommodate the crowds."

Santorini, formed after a volcanic eruption in 1613 B.C., is one of the world’s top tourist destinations and Greece’s most popular. It draws nearly two million people a year and almost half of them are day visitors from cruise ships that stay on the island for an average of seven hours, according to cruise operators and local officials. On the busiest days in the summer, Santorini draws more than 10,000 visitors a day.

The crowds are taking a toll, Mr. Zorzos said.

“A lot of people here depend on cruisers, but something has to give,” the mayor said in an interview. “The electricity grid and water supply are at their limit. Garbage has doubled in five years. If we don’t control the crowds, it will backfire and ruin us.”